L(s) = 1 | − 3-s + 5-s − 1.07·7-s + 9-s − 11-s − 4.34·13-s − 15-s + 7.75·17-s − 5.26·19-s + 1.07·21-s + 2.15·23-s + 25-s − 27-s + 1.41·29-s + 4.68·31-s + 33-s − 1.07·35-s − 2·37-s + 4.34·39-s − 9.41·41-s − 7.60·43-s + 45-s − 4.68·47-s − 5.83·49-s − 7.75·51-s + 0.156·53-s − 55-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.407·7-s + 0.333·9-s − 0.301·11-s − 1.20·13-s − 0.258·15-s + 1.88·17-s − 1.20·19-s + 0.235·21-s + 0.449·23-s + 0.200·25-s − 0.192·27-s + 0.263·29-s + 0.840·31-s + 0.174·33-s − 0.182·35-s − 0.328·37-s + 0.694·39-s − 1.47·41-s − 1.15·43-s + 0.149·45-s − 0.682·47-s − 0.833·49-s − 1.08·51-s + 0.0215·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 7 | \( 1 + 1.07T + 7T^{2} \) |
| 13 | \( 1 + 4.34T + 13T^{2} \) |
| 17 | \( 1 - 7.75T + 17T^{2} \) |
| 19 | \( 1 + 5.26T + 19T^{2} \) |
| 23 | \( 1 - 2.15T + 23T^{2} \) |
| 29 | \( 1 - 1.41T + 29T^{2} \) |
| 31 | \( 1 - 4.68T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 9.41T + 41T^{2} \) |
| 43 | \( 1 + 7.60T + 43T^{2} \) |
| 47 | \( 1 + 4.68T + 47T^{2} \) |
| 53 | \( 1 - 0.156T + 53T^{2} \) |
| 59 | \( 1 + 6.15T + 59T^{2} \) |
| 61 | \( 1 + 4.15T + 61T^{2} \) |
| 67 | \( 1 - 8.68T + 67T^{2} \) |
| 71 | \( 1 - 4.68T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 8.09T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.397429811661407650214068857193, −7.70840741781031099767292305126, −6.78014228439698119183105764037, −6.25295378954878238540627991351, −5.22847056274548011047854042672, −4.85183502210742771968797599572, −3.55370258268752626976725035274, −2.65870786203936261835002982436, −1.46515982737286110635773024591, 0,
1.46515982737286110635773024591, 2.65870786203936261835002982436, 3.55370258268752626976725035274, 4.85183502210742771968797599572, 5.22847056274548011047854042672, 6.25295378954878238540627991351, 6.78014228439698119183105764037, 7.70840741781031099767292305126, 8.397429811661407650214068857193